RATIOCINATION, OR SYLLOGISM. 229 



All valid ratiocination ; all reasoning by which, 

 from general propositions previously admitted, other 

 propositions equally or less general are inferred ; may 

 be exhibited in some of the above forms. The whole 

 of Euclid, for example, might be thrown without 

 difficulty into a series of syllogisms, regular in mode 

 and figure. 



Although a syllogism framed according to any of 

 these formulae is a valid argument, all correct ratioci- 

 nation admits of being stated in syllogisms of the 

 first figure alone. The rules for throwing an argu- 

 ment in any of the other figures into the first figure, 

 are called rules for the reduction of syllogisms. It is 

 done by the conversion of one or other, or both, of the 

 premisses. Thus, an argument in the first mode of 

 the second figure, as 



No C is B 



All A is B 



therefore 



No A is C, 



may be reduced as follows. The proposition, No C 

 is B, being an universal negative, admits of simple 

 conversion, and may be changed into No B is C, 

 which, as we showed, is the very same assertion in 

 other words the same fact differently expressed. This 

 transformation having been effected, the argument 

 assumes the following form : 



No B is C 



All A is B 



therefore 



No A is C, 



which is a good syllogism in the second mode of the 

 first figure. Again, an argument in the first mode of 

 the third figure must resemble the following : 



